Mathematical Sciences: Reciprocity for Fredholm Operators

数学科学：Fredholm 算子的互易性

• 批准号: 8702540
• 负责人: Richard Carey
• 金额: $ 4.05万
• 依托单位: University of Kentucky Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-06-15 至 1989-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8702540&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Reciprocity; Fredholm; Operators

Operator theory is a central discipline in Modern Analysis. Its origins lie in the study of mathematical physics and partial differential equations in the early twentieth century. At that time, it was seen that numerous physical problems in the theory of equilibria, vibration, quantum theory, etc. could be studied productively via the integral equations that model the phenomena. Thus, from the fertile minds of Hilbert, von Neumann, and other giants the subject of operator theory has grown to a central position in such investigations, and in core mathematics as well. At the heart of this methodology is the deep investigation of the spectrum of an operator. For self-adjoint or normal operators, this theory is now a standard technique throughout analysis, and the spectral theorem provides the necessary building blocks for all such operators. The current frontier, therefore, is the study of the structure of non-normal operators, as for example in the Fredholm theory. At this frontier, Professor Carey and his collaborator, Professor Pincus, are world leaders with established reputations for deep, conceptually innovative, fundamental, interdisciplinary work. Their theory of the principal function, defined on the spectrum of certain operators, is a broad extension of the ordinary index. This new research collaboration has established connections between the principal function and the distribution of zeroes of functions associated with ergodic flows, the structure of subnormal operators and the theory of closed currents in the study of function algebras. This work is particularly fertile as it directly combines operator theoretic ideas with other branches of mathematics, notably commutative ring theory and algebraic geometry. Currently, applications to the theory of commuting Toeplitz operators and linear and nonlinear partial differential operators are envisaged, among other technical applications in operator theory.

算子理论是现代分析中的一个中心学科。 它的起源在于研究数学物理和部分 微分方程在世纪早期。 在那个 当时，人们发现，理论中的许多物理问题 平衡，振动，量子理论等都可以研究 通过对现象进行建模的积分方程有效地进行。 因此，从希尔伯特、冯·诺依曼和 算子理论的主题已经发展成为一个 中心地位，在这样的调查，并在核心数学 也 这种方法的核心是深入调查 一个运营商的频谱。 对于自伴或正规 运营商，这个理论现在是一个标准的技术， 分析，谱定理提供了必要的 所有这些运营商的构建块。 目前的前沿， 因此，研究非正规算子的结构， 比如Fredholm理论。 在这个前沿， 凯里教授和他的合作者平卡斯教授是世界上 领导人与建立声誉的深刻，概念 创新、基础、跨学科的工作。 他们的理论 主函数，定义在某些 操作符，是普通索引的广泛扩展。 这个新 研究合作建立了联系， 主函数与函数零点的分布 与遍历流相关联，次正态的结构 算子和闭合电流理论在研究 函数代数 这项工作特别丰富，因为它 直接结合了算子理论的思想与其他分支， 数学，特别是交换环理论和代数 几何 目前，通勤理论的应用 Toeplitz算子与线性和非线性偏微分 在其他技术应用中， 算子理论